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In this project, we set out to develop constitutive models for the fluid-particle interaction force in fluid-particle mixtures.  Specifically, we were concerned with systems where different types of particle coexist in the mixture.  In such systems, the drag force exerted on a given type of particles by the fluid will depend not only on the relative (local-average) velocity between the fluid and that particle type, but also on the local average velocities of all the other particle types.  This complexity is not commonly recognized in models analyzing many practically relevant gas-particle systems such as fluidized bed reactors.

We set out to construct constitutive models for the fluid particle drag force in systems containing multiple types of particles.  Our approach can be summarized as follows:  we first created a configuration for the particle mixture that is consistent with hard sphere distribution by performing molecular dynamics-like simulations with the particles placed in a periodic box.  We then assigned velocities to each particle and forced a fluid flow through the interstitial region at a chosen local average value by applying a pressure drop.   The fluid flow simulations were done using the Lattice Boltzmann method.  After allowing the fluid flow to reach steady state, the force on each particle due to the fluid was computed.  These results were then averaged to compute the desired fluid-particle interaction force on each particle type due to the fluid.  Such simulations were repeated for many realizations of the particle microstructure to generate an ensemble average.

By repeating the above procedure for various phase velocities and various phase volume fractions, a large database on fluid-particle interaction force was created.  By analyzing this database, constitutive models were created.

We first analyzed the case of slow flow of fluid through the particle assembly, in the so-called Stokes flow regime where the fluid inertia plays a negligible role.  Here we first considered a bidisperse system of equally sized particles (which may differ, for example, in density or simply color).  The drag law constructed for this system is described in Yin & Sundaresan (Ind. Eng. Chem. Res., 48(1) 227-241 (2009)).  In this manuscript, we also showed how the constitutive model could be generalized for n-different types of particles.

We then considered Stokes flow of a bidisperse system where the particles were no longer assumed to be of the same size.  This dramatically increases the computational load as we now had to explore many different size ratios.    We successfully completed the development of analytical constitutive models for bidisperse systems, and also demonstrated how it can be generalized for systems with many different types of particles.  This work is described in Yin & Sundaresan (AICHE J., 55(6) 1352-1368 (2009)).

We have embarked on a study to extend the above results to the case when fluid inertia is no longer unimportant.  In other words, the Reynolds number based on the relative velocity between a given particle type and the fluid, fluid density, fluid viscosity and the particle diameter is no longer small compared to unity.  We considered Reynolds number up to 40 and generated computational data on the fluid-particle interaction force in bidisperse systems.  We have identified how the constitutive models valid for Stokes flow conditions (described in Yin & Sundaresan, AICHE J., 55(6) 1352-1368 (2009)) can be modified to account for the inertial effects.  A manuscript based on this work is under review.